Become familiar with probability terms,
symbols, and properties. For as many as possible and
appropriate, you should have at least one example to help
explain and clarify the term, symbol, or property.

So that we can be efficient and clear in our discussions and
calculations associated with probability, we will identify some
terminology and symbolism to help us. We also point out some
fundamental properties of probability.

outcomes: the possible results of an experiment

equally likely outcomes: a set of outcomes that each
have the same likelihood of occurring.

nonuniform sample space: a sample space that contains
two or more outcomes that are not equally likely.

event: a collection of one or more elements from a
sample space.

expected value: the long-run average
value of the outcome of a probabilitic situation; if an
experiment has n outcomes with values a(1), a(2), . . . , a(n),
with associated probabilities p(1), p(2), . . . , p(n), then
the expected value of the experiment is

random event: an experimental event that has no
outside factors or conditions imposed upon it.

P(A): represents the probability P for some event
A.

probability limits: For any event A, it must be that
P(A) is between 0 and 1 inclusive.

probabilities of certain or impossible events: An
event B certain to occur has P(B) = 1, and an event C that is
impossible has P(C) = 0.

complementary events: two events whose probabilities
sum to 1 and that share no common outcomes. If X and Y are
complementary events, that P(A) + P(B) = 1.

mutually exclusive events: two events that share no
outsomes. If events C and D are mutually exclusive, then P(C or
D) = P(C) + P(D); if two events are not mutually exclusive,
then P(C or D) = P(C) + P(D) - P(C and D).

independent events: two events whose outsomes have no
influence on each other. If E and F are independent events,
than P(E and F) = P(E) * P(F).

conditional probability: the determination of the
probability of an event taking into account that some condition
may affect the outcomes to be considered. The symbol P(A|B)
represents the conditional probability of event A given that
event B has occurred. Conditional probability is calculated as
P(A|B) = P(A and B)/P(B).

geometrical probability: the determination of
probability based on the use of a 1-, 2-, or 3-dimensional
geometric model.

Expected Value

Expected value is the long-run average value of the outcome of
a probabilitic situation. If an experiment has n outcomes with
values a(1), a(2), . . . , a(n), with associated probabilities
p(1), p(2), . . . , p(n), then the expected value of the
experiment is a(1)*p(1)+ a(2)*p(2) + . . . + a(n)*p(n).

For example, if the experiment is to roll a fair die and record
the output showing on the top face of the die, the outcomes are
{1,2,3,4,5,6}. Because each outcome is equally likely, each
outcome has probability of 1/6. The expected value for this
experiment is the weighted average of the outcomes and the
probabilities, calculated using the summation shown above:

The expected value provides us information about what to expect
is the experiment is carried out many many times. Here, the
average output we will get is 3.5. Note that 3.5 is not one of the
possible outcomes of the experiment, but rather a weighted
average, representing what a "typical" outcome will be over the
long-term running of the experiment.

Expected value is a useful concept to turn to when evaluating
whether a game of chance is fair. Suppose that Allen and Zenda
play a game with a die, each rolling the die on alternate turns.
When someone rolls a 5 or 6, Allen wins as many points as are
shown on the die. When someone rolls a 1,2,3, or 4, Zenda wins as
many points as are showing on the die. The first player to
accumulate 25 points or more is the winner. What is the expected
value for each player?

For Allen, the expected value is (5)*(1/6) + (6)*(1/6) = 11/6.
For Zenda, it is (1)*(1/6) + (2)*(1/6) + (3)*(1/6) + (4)*(1/6) =
10/6. This is not a fair game, for in the long run, Allen will win
more points per play than Zenda.

Here's another example.

There are four possible outcomes when a gambler plays
the spinner game shown to the right. The green area, $50,
represents 1/2 the circle, the violet area, $20,
represents 1/3 of the circle, and each of the other areas
($10 and $0) is 1/12 of the circle.

A player spins the spinner and wins the amount upon
which the spinner lands.

What should the player pay so that this is a fair
game?

The expected winnings are

(50)(1/2) + (20)(1/3) + (10)(1/12) + (0)(1/12), which
equals $32.50.

If that is what a player can expect to win per play,
over the long haul, then to make this a fair game the
player should pay $32.50 per spin. Any less than that and
the player has an advantage. Any more than that and the
game owner will earn a profit, again, over many many
plays of the game.

Before you go to Las Vegas to win your fortune, be
sure to calculate the expected value of the games you'll
be playing!